Longest weakly increasing subsequences of discrete random walks on the integers with heavy tailed distribution of increments
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910087908425728 |
|---|---|
| author | Mendonça, José Ricardo G. Freire, Marcelo V. |
| author_facet | Mendonça, José Ricardo G. Freire, Marcelo V. |
| contents | We investigate the behavior of the length of the longest weakly increasing subsequences (weak LIS) of $n$-step random walks with nonzero integer increments $k = \pm 1, \pm 2, \dots$ given by a zero-mean, symmetric heavy tailed mass distribution proportional to $|k|^{-1-α}$ for several values of the real parameter $α> 0$ together with that of the simple random walk ($k=\pm 1$), to which the $n$-step heavy tailed random walks tends when $α> (1+o(1))\log_{2}{n}$.
By means of exploratory fits, weighted nonlinear least squares, and ANOVA model comparison, we found that the sample average length $\langle{L_{n}}\rangle$ scales like $\langle{L_{n}}\rangle \sim \sqrt{n}\log{n}$ when the distribution of increments has finite variance ($α> 2$) and $\langle{L_{n}}\rangle \sim n^θ$ with a varying exponent $θ> 0.5$ when the variance is infinite ($α\leq 2$). Distributional diagnostics indicate that the bulk of the $L_{n}$ distribution is very well-approximated by a lognormal model, though systematic deviations are observed in the tails.
Our results corroborate and expand upon previous results for the LIS of other types of heavy-tailed random walks and raise a conjecture as to whether the distribution of $L_{n}$ is given, or can be effectively described, by a lognormal distribution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29047 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Longest weakly increasing subsequences of discrete random walks on the integers with heavy tailed distribution of increments Mendonça, José Ricardo G. Freire, Marcelo V. Statistical Mechanics Probability Applications We investigate the behavior of the length of the longest weakly increasing subsequences (weak LIS) of $n$-step random walks with nonzero integer increments $k = \pm 1, \pm 2, \dots$ given by a zero-mean, symmetric heavy tailed mass distribution proportional to $|k|^{-1-α}$ for several values of the real parameter $α> 0$ together with that of the simple random walk ($k=\pm 1$), to which the $n$-step heavy tailed random walks tends when $α> (1+o(1))\log_{2}{n}$. By means of exploratory fits, weighted nonlinear least squares, and ANOVA model comparison, we found that the sample average length $\langle{L_{n}}\rangle$ scales like $\langle{L_{n}}\rangle \sim \sqrt{n}\log{n}$ when the distribution of increments has finite variance ($α> 2$) and $\langle{L_{n}}\rangle \sim n^θ$ with a varying exponent $θ> 0.5$ when the variance is infinite ($α\leq 2$). Distributional diagnostics indicate that the bulk of the $L_{n}$ distribution is very well-approximated by a lognormal model, though systematic deviations are observed in the tails. Our results corroborate and expand upon previous results for the LIS of other types of heavy-tailed random walks and raise a conjecture as to whether the distribution of $L_{n}$ is given, or can be effectively described, by a lognormal distribution. |
| title | Longest weakly increasing subsequences of discrete random walks on the integers with heavy tailed distribution of increments |
| topic | Statistical Mechanics Probability Applications |
| url | https://arxiv.org/abs/2603.29047 |